To get back to your original question, I do not see any problem with applying ED to Williams as long as (i) You are sure it is your top choice (this seems solid based on what you said); (ii) You are pretty sure that you can afford it (either you are fine being full pay, or you have run the NPC and you and your parents are fine with the results); and (iii) You have also figured out and applied to at least two solid safeties.
So, I think that you need to check the full price at Williams, plan for 5% per year inflation, and see if your parents are fine with this. If not, then you need to run the NPC. You will need help from your parents to run the NPC. If your parents own a small business, farm, or rental property or of your parents are divorced then the NPCs will be way too optimistic.
Then you need to find two solid safeties that you know you will get accepted to, you know you can afford, and that you would be happy attending. I have not seen any safety mentioned so far on this thread.